On sound production in brass instruments

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The brass player's lips under playing conditions are displaced both longitudinally along the axis of the air column entryway, and laterally, or perpendicular, to the axis. Existing theories have modeled the valving action of the player's lips as a simple oscillator, or lip reed, but fall into two classes, depending upon whether the lip reed displacement is assumed to be purely longitudinal or lateral. An alternative model of sound production is proposed, retaining the single degree-of-freedom oscillator description, but including both longitudinal and lateral displacements. The lip reed is assumed to execute a circular, outwardly rolling motion. In the limit of small oscillations, this simplifies to linear displacement along an axis inclined relative to the air column entryway axis. The angle of inclination is a model parameter, but should depend upon mouthpiece geometry and player's embouchure. The lip reed is driven longitudinally by the pressure difference between the player's mouth (upstream) and the mouthpiece (downstream). It is driven laterally by the pressure within the lip orifice, which is related to the downstream pressure by momentum flux conservation. The time domain model incorporates reflection functions looking downstream into the air column, and upstream into the player's respiratory tract. The upstream reflection function can similarly be incorporated into player-woodwind models.